Chaos Theory MS Electrical Engineering Department of Engineering

Chaos Theory MS Electrical Engineering Department of Engineering

Chaos Theory MS Electrical Engineering Department of Engineering GC University Lahore Course Contents

Introduction Flows on the line Bifurcations Flows on the circle Linear Systems Phase Planes Limit Cycles Lorenz Equation One dimensional Maps

Fractals Strange Attractors Books NONLINEAR DYNAMICS AND CHAOS With Applications to Physics, Biology, Chemistry, and Engineering, STEVEN H. STROGATZ CHAOS AND NONLINEAR DYNAMICS: An Introduction for Scientists and Engineers,

Robert C. Hilborn The illustrated Dictionary of NONLINEAR DYNAMICS AND CHAOS, Tomasz Kapitaniak, Steven R. Bishop Research Chaos, An Interdisciplinary Journal of Nonlinear Science International Journal of Bifurcation and

Chaos Chaos, Solitons and Fractals Chaos Meaning Pronunciation:/kes/noun complete disorder and confusion Example: snow caused chaos in the region Physics: the property of a complex system whose behaviour is

so unpredictable as to appear random, owing to great sensitivity to small changes in conditions. the formless matter supposed to have existed before the creation of the universe.

Reference: Oxford Dictionary Probabilistic vs Deterministic Random Deterministic

Static vs Dynamic Static Systems Dynamic Systems

Dynamics: Subject that deals with change systems that evolve in time, settles down to equilibrium, keeps repeating in cycles, or does something more complicated Brief History of Dynamics 17th Century Newton solving Two-Body Problem (Sun & Earth) using Differential Equations and Law of Gravitation

Three-Body Problem (Sun, Moon and Earth) N o explicit solution Late 19th Century Poincare Qualitative Solution rather than Quantitative, Geometric Approach Dynamics restricted to Nonlinear oscillators in radio, radar, phase-locked loops, and lasers Lorenz's discovery in 1963 of chaotic motion on a strange attractor Weather Model: Aperiodic, very sensitive to Initial Conditions

Types of Dynamical Systems Differential equations: the evolution of systems in continuous time, whereas iterated maps arise in problems Iterated maps (Difference equations): time is discrete

Differential Equations LINEAR Differential Equations Exponential Growth of population of organisms

Differential Equations NONLINEAR Trajectory & Phase-Space Draw the trajectories without actually solving the system Non-autonomous

Systems Time Dependent Systems Dimension of the Phase Space n = 1: Growth, Decay or Equilibrium e.g. RC Circuit (Linear), Logistic Equation (Nonlinear) n = 2: Oscillations e.g RLC Circuit

(Linear), Pendulum (Nonlinear) n >= 3: Three-Body Problem, Chaos & Fractals (Nonlinear) Example: Example (Continued) Application Areas

Mathematics Biology Computer science Economics Engineering Finance

Philosophy Physics Politics Population dynamics Psychology Chaos in Electrical Circuits

Chaos in Electrical Circuits Secure Communication

Recently Viewed Presentations

  • Ethics: An Introduction

    Ethics: An Introduction

    Ethics: The Science of Morals. Ethics is itself broken into numerous categories: Meta-ethics. the theoretical inquiry of moral principles and how their truth values (if any) may be determined. Normative ethics. Concerning the practical means of determining a moral course...
  • Product Evaluation & Quality Improvement

    Product Evaluation & Quality Improvement

    IF YOU DID NOT COMPLETE THE PRE-LAB, there are extra fixed robots you can use for the duration of the lab period. ... Report. Individual lab report. Title page. Discussion topics in the manual. MUST. include spreadsheet with test results,...
  • Parent Session Introduction Australian Curriculum

    Parent Session Introduction Australian Curriculum

    -Personal and Social capability-Intercultural understanding-Ethical understanding. The general capabilities play a significant role in the Australian Curriculum in equipping young Australians to live and work successfully in the twenty-first century. In the Australian Curriculum, capability encompasses knowledge, skills, behaviours and...
  • WHOOOA BIG SAFETY ISSUE! 1 Scientifi c Method

    WHOOOA BIG SAFETY ISSUE! 1 Scientifi c Method

    Just as it does for a professional scientist, the scientific method will help you to understand what makes a scientific experimental inquiry valid. 3 But first let us review the steps of the Sci. Meth."Complete supportive class notes"
  • CPI 360: Decision Making and Problem Solving The

    CPI 360: Decision Making and Problem Solving The

    Original slide numbers given. Add one to each to offset for this slide. ... Personnel Influencing problem and choice definition. ... How do Tech., Culture & Ethics influence Decision Making? Fuzzy Logic and Neural Networks.
  • Computer Modeling Fundamentals - PC\|MAC

    Computer Modeling Fundamentals - PC\|MAC

    Graphics Window. The active modeling area where parts and assemblies are created and edited. Quick Access Toolbar. Graphics. Window. Application Menu. Ribbon. Computer Modeling Fundamentals. PLTW Gateway® Unit 1 - Lesson 1.5 - Designing For Production. Show students how to...
  • Native Dynamic SQL Exposed - Amazon S3

    Native Dynamic SQL Exposed - Amazon S3

    SQL> SELECT column_value AS "Three Stooges"2 FROM TABLE(sql_varray('Moe','Larry','Curly')); Casts the SQL varray into a SQL consumable result set. ... Update the nested ADT collection by finding and replacing the 'Shemp' value with 'Curly'. Type of Oracle ObjectsLists or TABLE data...
  • How Does National Life Group Make Money?

    How Does National Life Group Make Money?

    life events happen; some expected and some unexpected. The life events you . need to prepare . for. Life Happens What if you died too soon? Life Insurance provides death benefit protection and helps to provide security for your family.